When you click the button, it seems as if it is downloading, there is a progress bar near
the URL bar but when it finishes, you are left on the download page and the addon is not
listed on the browser's addons page.

I also can't download it and I don't have a Galaxy S8 so it has nothing to do with that.

I do have FF 53 though so I think it might be that. I had NoScript on this device running fine but I decided to uninstall it and reinstall it for some reason, and now I can't reinstall it. I click the download link as you describe and it looks like it is trying but then it does nothing and just sits there.
So far the only thing I can come up with is that this new version of FF is blocking the install somehow. Odd though that if it is already installed when you update to 53 it does not remove it. So for now the only workaround I can think of would be to try uninstalling FF, installing an older version of FF, installing NS, then upgrading FF back up to 53. At this point I'm not wiling to try that yet.

So if you have FF 53 with NS installed DON'T UNINSTALL IT. You won't be able to get it back at this point.

Does anyone have a better link to install NS that might work? The button does not.

I have the same problem on an up to date Galaxy Tab S2 ( Android 6.0.1 ), using Firefox 52.0.1 ( when I check for updates in FF, this is the latest available ) . I cannot install Noscript 5.0.3 ( or 5.0.2 ). The closest I get is that Noscript appears in the list of addons, but is greyed out and there is no toolbar icon. During installation I see a dialog saying my addon is installed and I can access it from the toolbar, but all I can do then is remove it.

Ah, sorry, I jumped tracks mid comment. When I try to install the nsa NoScript, then I do see the same behaviour as the op ( nothing listed in the add on list ). When I install the 'normal' 5.0.3, things appear to work better as described ( noscript appears to install and can be uninstalled ), but actually noscript still isn't working.